Cyclic Causal Model Discovery in Psychology

Network Theory in Psychology

Mental disorder is produced by direct causal interactions between symptoms that reinforce each other via feedback loops. (Borsboom & Cramer, 2013).

Inferring causality from observational data

And what has happened a lot in psychology is that many empirical researchers try to gain insights into this causal relationships by fitting this statistical network model using observational data. [Alt + click!] But these connections in the network are merely statistical relations, which cannot be directly translated into causal relations.

And more and more people have become aware of that, and there has come another popular modeling tool, which is DAG. DAG stands for directed acyclic graphical model. And it is specifically designed to make causal inference based on the observational data.[Alt + click!] It has all these arrows you can interpret as a cause and effect. It seems very promising, however, one of biggest shortcomings of DAG is by definition it does not allow cycles. But as we saw before, the true system contains cycles. In that sense, we cannot really use DAG to estimate the real underlying mechanism. So what do we need to do?

Directed Acyclic Graph (DAG)

Directed Cyclic Graph (DCG)

Problem: Estimating a cyclic causal model is fundamentally very difficult. Relaxing the acyclicity assumption entails much of theoretical complication.

Solutions exist

Cyclic Causal Discovery (CCD) (Richardson, 1996)
Fast Causal Inference (FCI) (M. Mooij & Claassen, 2020)
Cyclic Causal Inference (CCI) (Strobl, 2019)

Solutions exist, but…

PAG: Partial Ancestral Graph ?
PAAG: Partially-oriented MAAG ?
MAAG: Maximal Almost Ancestral Graph ?

Project goals

Give an accessible overview of the algorithms
Investigate the performance of each algorithm
Apply to the empirical data

Example

Summary

Causal inference is the fundamental interest in science.
The underlying dynamic processes of many systems contain cycles.
Learning cyclic causal models from observational data is challenging.
Our study will showcase the cyclic causal discovery algorithms that are potentially suitable for typical psychological observational data.

References

Borsboom, D., & Cramer, A. O. J. (2013). Network analysis: An integrative approach to the structure of psychopathology. Annual Review of Clinical Psychology, 9(1), 91–121. https://doi.org/10.1146/annurev-clinpsy-050212-185608

M. Mooij, J., & Claassen, T. (2020). Constraint-based causal discovery using partial ancestral graphs in the presence of cycles. In J. Peters & D. Sontag (Eds.), Proceedings of the 36th conference on uncertainty in artificial intelligence (UAI) (Vol. 124, pp. 1159–1168). PMLR. https://proceedings.mlr.press/v124/m-mooij20a.html

Richardson, T. (1996). Discovering cyclic causal structure. Carnegie Mellon [Department of Philosophy].

Strobl, E. V. (2019). A constraint-based algorithm for causal discovery with cycles, latent variables and selection bias. International Journal of Data Science and Analytics, 8(1), 33–56. https://doi.org/10.1007/s41060-018-0158-2

Evaluation metrics

Precision = \(\frac{TP} {(TP + FP)} = \frac{a}{(a + d + g)}\)
Recall = \(\frac{TP} {(TP + FN)} = \frac{a}{(a + b + c)}\)
Uncertainty rate = \(\frac{\text{Number of circle endpoints} (\circ)}{\text{Total number of possible endpoints}}\)

Simulation settings

Theoretical Complications

Global Markov property is no longer guaranteed.
- Need extra restrictions on \(P\) (e.g., linearly independent error terms (\(\varepsilon\)))
Cyclic model is not always statistically identified (even in linear case).
- many equivalent models !
Equilibrium state is necessary (All \(|\lambda| < 1\)).

There are many problems with assumptions when estimating causal model with cycles. But I picked three the most cruicial ones. So when it comes to cyclic model, the most crucial issues are first, the global markov property which is the necessary property in order to read off the graphical model, that is often violated in the models with cycles.
And another major issue is that the cyclic models are often not statistically identified, meaning that either there are too many solutions possible or no solution at all.
Lastly, not all cycles can be estimated. The ones we can estimate are limited such that it has to converge to some equilibrium states, and in order for that to happen, all eigen values of regression matrix has to be smaller than 1!

So this causes quite some headaches when trying to estimate the cyclic models.

Constraint-Based Algorithms

Possible Practical Application

Personalized psychotherapy (target symptoms)
Medical: effective treatment design

Q:“What is the expected benefits/applications if you actually succeed in estimating cyclic model?”

For example in psychology, the possible application would be that we can customize therapy based on the found causal dynamics. For example, imagine that this is a person’s real depression model, then we can see that insomnia is the central node that has a lot of connections causing other depression symptoms. So we can effectively treat this patient’s depression by targeting the sleep problems.
I think this would also apply more or less the same way to other medical treatments. You can imagine that this is a causal model for some other disease, right. Then, we can also design a more effective treatment once we identify which is the most influential cause of the disease. So yeah this would be one of the crucial practical implications of this project.

Follow-up

Possible combination with different types of causal discovery algorithm. \(\rightarrow\) Hybrid!

CCD+ GES (greedy equivalence search)

Q:“What could be an interesting follow-up or extension you could do if more fundings become available?”

If there are more fundings available, one of the interesting extension to this could be coming up with a hybrid algorithm, such that combining different types of algorithms together. Because one of the general limitations of the causal discovery algorithms is that they often do not give you the full picture of the causal system, meaning that there are always some directions that the algorithm cannot correctly identify.

So for example let’s say this is the true model. And usually, the algorithms can only identify up to a certain extent. So you see these edges are left undirected. Then we can use another algorithm, to orient the rest of edges.

For example, we could combine CCD with GES, greedy equivalence search algorithm, which utilizes different types of information to find an optimal model.

So like this, we can combine these two and get the full picture.

This extension is very plausible in my opinion and promising but it’s not known yet, which combination would work the best in practice. So this would be a great work to do if there is extra resources available in the future.

Thank you